Proof:    Notice that \(\mathcal {G}^{\prime }\)  is a subgraph of \(\mathcal {G}\) , and hence \(\widehat{\mathcal {G}^{\prime }}\)  is a subcomplex in \(\widehat{\mathcal {G}}\) .
Let \(u\)  be dominated by \(v\)  in \(\mathcal {G}\) . Then, we can write a deformation retract from \(\widehat{\mathcal {G}}\)  to \(\widehat{\mathcal {G}}^{\prime }\)  by pushing the edge \(e_{uv}\)  starting from \(u\)  toward \(v\) . In other words, by using the simplicial coordinates, one can define a homotopy \(F:\widehat{\mathcal {G}}\times I\rightarrow \widehat{\mathcal {G}}\)  which is identity on \(\widehat{\mathcal {G}}^{\prime }\)  and pushing all the faces in \(\widehat{\mathcal {G}}-\widehat{\mathcal {G}}^{\prime }\)  to the corresponding faces in \(\widehat{\mathcal {G}}^{\prime }\) . This gives a homotopy equivalence \(\widehat{\mathcal {G}}\sim \widehat{\mathcal {G}}^{\prime }\) . To visualize, in Figure REF , one can push vertex 1 in the clique complex \(\widehat{\mathcal {G}}\)  towards vertex 3 along the edge between them. After the push, the 2-simplices [1,2,3] and [1,3,4] are pushed to the edges [2,3] and [3,4] respectively. See [1]}, [2]} and [3]} for details.
\(\square \)
